@inproceedings{Tal2018,
title = {Continuous Tensor Train-Based Dynamic Programming for High-Dimensional Zero-Sum Differential Games},
author = {Ezra Tal and Alex Gorodetsky and Sertac Karaman},
url = {http://www.alexgorodetsky.com/wp-content/uploads/2018/03/tal-ea-ACC2018.pdf},
year = {2018},
date = {2018-06-28},
booktitle = {American Control Conference (ACC)},
address = {Milwaukee, WI},
abstract = {Zero-sum differential games are a prominent research topic in a wide variety of fields ranging from motion planning under adversarial conditions to economics modeling. In the field of robust control, synthesis using game theory can also be used to obtain performance guarantees. However, most existing computational methods for differential games suffer from the curse of dimensionality, i.e., the computational requirements grow exponentially with the dimensionality of the state space. In the present work, we aim to alleviate the curse of dimensionality with a novel dynamic-programming-based algorithm that uses a continuous tensor-train approximation to represent the value function. This approximation method can represent and compute with high-dimensional tensors using computational resources that grow polynomially with dimensionality and polynomially with the rank of the value function. The proposed algorithm is shown to converge to optimal solutions with arbitrary bounds for a general class of differential games. The new algorithm is demonstrated using several game scenarios, achieving more than five orders of magnitude savings in memory and three order of magnitude in computational cost in a six-dimensional problem, when compared to the standard value iteration.},
keywords = {Differential Games, Stochastic Optimal Control, Tensor decompositions},
pubstate = {forthcoming},
tppubtype = {inproceedings}
}

Zero-sum differential games are a prominent research topic in a wide variety of fields ranging from motion planning under adversarial conditions to economics modeling. In the field of robust control, synthesis using game theory can also be used to obtain performance guarantees. However, most existing computational methods for differential games suffer from the curse of dimensionality, i.e., the computational requirements grow exponentially with the dimensionality of the state space. In the present work, we aim to alleviate the curse of dimensionality with a novel dynamic-programming-based algorithm that uses a continuous tensor-train approximation to represent the value function. This approximation method can represent and compute with high-dimensional tensors using computational resources that grow polynomially with dimensionality and polynomially with the rank of the value function. The proposed algorithm is shown to converge to optimal solutions with arbitrary bounds for a general class of differential games. The new algorithm is demonstrated using several game scenarios, achieving more than five orders of magnitude savings in memory and three order of magnitude in computational cost in a six-dimensional problem, when compared to the standard value iteration.